Exact and Approximate Expressions for Bubble–Particle Collision

An Approximate Analytical Expression for the Probability of Attachment by Sliding

“…In actuality, as [8]; however, the second term on the right-hand side of this equation has been dropped in view of assumption A2, above. In [8], v ps represents the particle settling velocity given by v ps ϭ ṽ ps [9] and…”

“…In fact, if v ps ϭ 0 then, by virtue of [8], v p rel ϭ u , where, for an assumed intermediate flow over the bubble surface, it follows from the work of Yoon and Luttrell (7) that…”

“…In deriving an expression for P asl in this section we will not need to make use of a force balance equation for the sliding particle in the tangential direction; as will be seen in the analysis to follow, a judicious use of [8], in combination with the appropriate form of the force balance equation in the radial direction, suffices to produce the desired approximate analytical expression for P asl .…”

“…Indeed, it is believed by the authors that the expression for P asl which is derived here represents the first, analytical, closed-form (albeit, approximate) formula for this key flotation microprocess probability that has appeared in the literature. This model for P asl will be incorporated into an overall model of the flotation separation process currently being developed (1)(2)(3)8).…”

“…(a) the approach of a particle to an air bubble with the subsequent collision with, or interception of the particle by, the bubble (for particles the size of typical ink particles found in flotation deinking, the main focus here is on the zone of possible interaction which is created when the particle approaches a sufficiently small distance from the bubble); for recent progress on this aspect of the problem we refer the reader to (8) and the discussions contained therein; (b) the formation of a three-phase contact angle after sliding of the particle along the thin liquid film which separates the particle from the bubble and the subsequent thinning and rupture of this film; and (c) the stabilization of the bubble/particle aggregate and its transport to the froth layer for removal.…”